Quote: any progression matches Peano's five axioms for numbers; so the axioms do not define '0', 'number' and 'successor'

Quotation

(3) Let "0" mean the number one, let "number" mean the set 1, 1/2, 1/4, 1/8, 1/16, ... and let "successor" mean "half". Then all Peano's five axioms will be true of this set. ... In fact, given any series x_0, x_1, x_2, x_3, ...x_n, ... [(a progression)] which is endless, contains no repetitions, has a beginning, and has no terms that cannot be reached from the beginning in a finite number of steps, we have a set of terms verifying Peano's axioms. ... [p. 8] It can be proved, conversely, that every series which verifies Peano's five axioms is a progression. ... [p. 9] This point, that "0" and "number" and "successor" cannot be defined by means of Peano's five axioms, and must be independently understood, is important. We want our numbers not merely to verify mathematical formulae, but to apply in the right way to common objects.   Google-1   Google-2

Published before 1923

Additional Titles

Quote: the concept of number should apply in the right way to common objects

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